A regular determinant of binomial coefficients

Abstract

Abstract.Let n be a positive integer and suppose that each of {a"}J and (cs}J is an increasing sequence of nonnegative integers. Let M be the nxn matrix such that Afij=C(ai, c,), where C(m, ri) is the number of combinations of m objects taken nata time. We give an explicit formula for the determinant of M as a sum of nonnegative quantities. Further, if a,g:c,, ¡ = 1, 2, ■ ■ ■ , n, we show that the determinant of M is positive.

A determinant of the Artin-Hasse exponential coefficients

“…To prove Theorem 1.1, we reduce the determinant |u pi−j | 1≤i,j≤ℓ to one involving only binomials. Then, using a result of Tonne [7], we are able to explicitly compute the determinant by counting certain types of Young tableaux.…”